Interval-valued SHAP in Tree-Based Models

Chenrui Zhu; Vu-Linh Nguyen; Marie-Hélène Masson; Sébastien Destercke

cs.LGOct 8, 2026

Interval-valued SHAP in Tree-Based Models

Authors: Chenrui Zhu, Vu-Linh Nguyen, Marie-Hélène Masson, Sébastien Destercke

Organizations: Université de technologie de Compiègne, UMR-CNRS 7253 Heudiasyc Compiègne, France · IUT de l’Oise, Université de Picardie Jules Verne Beauvais, France

Abstract

Shapley values are among the most popular feature-attribution explanations. Efficient approaches for computing/estimating Shapley values for tree-based models, which are state-of-the-art for tabular data sets, have been developed. However, it is known that Shapley values can be (highly) unrobust due to small and realistic changes. In this paper, we propose an imprecise Dirichlet model (IDM) based method to analyze the robustness of Shapley values in decision trees and random forests. Technically, it is done by quantifying and analyzing the interval-valued Shapley values when a few unannotated instances are randomly introduced to the leaves of the trees. The interval-valued Shapley values can be defined following common principles in handling incomplete data: the pessimistic and averaging principles. We derive various theoretical results that lead to efficient computation of the interval-valued Shapley values. We also show that the proposed method can be straightforwardly generalized to the case of Banzhaf values. We then present various case studies and experiments to illustrate the behaviour of the proposed interval-valued Shapley values and their applications in debiasing uninformative features.

Figures & tables

leaf	$n_{\ell}$	$N_{\ell}$	$w_{\ell}$	$m_{\ell}$	$[\underline{w}_{\ell},\overline{w}_{\ell}]$
$\ell_{1}$	20	60	$1/3$	1	$\left[\frac{20}{61},\frac{21}{61}\right]$
$\ell_{2}$	20	50	$2/5$	2	$\left[\frac{20}{52},\frac{22}{52}\right]$
$\ell_{3}$	10	30	$1/3$	1	$\left[\frac{10}{31},\frac{11}{31}\right]$
$\ell_{4}$	30	50	$3/5$	2	$\left[\frac{30}{52},\frac{32}{52}\right]$

TABLE I : IDM leaf-output intervals.

	$\displaystyle\Delta_{\ell}(m_{\ell}+1)$	$\displaystyle:=\alpha^{-}_{i,\ell}\left(\frac{m_{\ell}+1}{N_{\ell}+m_{\ell}+1}-\frac{m_{\ell}}{N_{\ell}+m_{\ell}}\right)$
		$\displaystyle=\alpha^{-}_{i,\ell}\frac{N_{\ell}}{(N_{\ell}+m_{\ell}+1)(N_{\ell}+m_{\ell})};$		(19)

Algorithm 1 Optimal allocation of virtual samples

step	leaf	$\Delta_{\ell}$	$m$
1	$\ell_{1}$	0.00984	$(1,0,0,0)$
2	$\ell_{4}$	0.00980	$(1,0,0,1)$
3	$\ell_{3}$	0.00968	$(1,0,1,1)$
4	$\ell_{1}$	0.00952	$(2,0,1,1)$
5	$\ell_{4}$	0.00943	$(2,0,1,2)$
6	$\ell_{1}$	0.00922	$(3,0,1,2)$

TABLE II : Worst-case and stochastic allocation computations.

	Diab.	BreC.	Ions.	Mort.
Features	8	30	34	79
Instances	768	569	351	14264

TABLE III : Summary of datasets

TABLE IV : Representative instances from the Diabetes dataset, their nominal SHAP values, and magnitude-based intervals.

(+)

and

(-)

indicate support for and opposition to the diabetic prediction, respectively

Table 6

Fig. 1 : Comparison of (row) nominal SHAP, $\mathrm{SHAP}^{\mathrm{Pess}}$ , $\mathrm{SHAP}^{\mathrm{Shrunk}}$ , and $\mathrm{SHAP}^{\mathrm{oob}}$ , where only $X_{2}$ is informative, for different signal strengths (column) $\rho\in[0,0.3]$ .

SHAP	$\mathrm{SHAP}^{\mathrm{Safe}}$	$\mathrm{SHAP}^{\mathrm{Pess}}$	$\mathrm{SHAP}^{\mathrm{Shrunk}}$	$\mathrm{SHAP}^{\mathrm{oob}}$
0.82	0.81	0.83	0.79	0.85

TABLE VII : Average AUC scores for separating informative from uninformative features.

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Shapley values are a standard tool for explaining predictions of tree ensembles, with Path-Dependent SHAP being the most widely used variant. Despite substantial progress, existing methods still exhibit trade-offs between depth-dependent runtime, numerical stability, and support for higher-order interactions. To address these challenges, we introduce Quadrature-TreeSHAP, a quadrature-based reformulation of Path-Dependent TreeSHAP that is numerically stable, naturally extends to any-order Shapley interaction values and is practically insensitive to tree depth. Our implementation supports both CPU and GPU and is integrated into XGBoost. Our method is based on a weighted-Banzhaf interaction polynomial, which expresses Banzhaf interaction values as expectations under a feature participation probability

p

. Shapley values and any-order interaction values are then recovered by integrating these polynomials over

p

from 0 to 1. We evaluate these integrals using Gauss-Legendre quadrature, and show that, in practice, only 8 fixed quadrature points are sufficient to reach machine precision. In fact, Quadrature-TreeSHAP with 8 fixed points achieves greater numerical stability than TreeSHAP. This fixed-point formulation removes depth dependence from the inner computation and enables efficient SIMD execution. We confirm these advantages empirically. On 12 XGBoost benchmarks, Quadrature-TreeSHAP computes Shapley values 1.06x-10.59x faster than TreeSHAP on CPU and 1.84x-6.95x faster than GPUTreeSHAP on GPU. Shapley pairwise interactions are 3.80x-58.11x faster on CPU, with higher-order interactions achieving speedups of up to 1200x compared to TreeSHAP-IQ.

Abstract

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